So we see that $\displaystyle f^{-1}(B) $ IS A subset of X. Now since for any subset ,lets say A, of X we have by definition f(A) ={ f(x) : xεA } then $\displaystyle f( f^{-1}(B))$ ={ f(x): xε$\displaystyle f^{-1}(B)$ }................................................. ...........2

So we see that $\displaystyle f( f^{-1}(B))$ is a set of the images of x f(x) , such that xε$\displaystyle f^{-1}(B)$.